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Fractional $Q$-curvature on the sphere and optimal partitions

Héctor A. Chang-Lara, Juan Carlos Fernández, Alberto Saldaña

TL;DR

This work studies a fractional Yamabe-type problem on the sphere, where the cost is governed by the fractional Q-curvature via the conformal fractional Laplacian $\mathscr{P}_g^s$. The authors exploit $G=O(m)\times O(n)$-symmetry to recover compactness in a critical nonlocal setting and prove the existence of a symmetric minimal partition of the sphere, with the partition profiles arising as phase-separated limits of a weakly coupled fractional system. A key methodological innovation is a self-contained, symmetry-driven reduction to a one-dimensional setting using Jacobi polynomials, which yields a new Hölder regularity result for symmetric $H_g^s$-functions on the sphere (for $s>\tfrac12$). The work further demonstrates the existence of infinitely many symmetric, fully nontrivial solutions to nonlocal competitive systems on the sphere and in Euclidean space, together with a precise description of the asymptotic phase-separation behavior as coupling tends to minus infinity, and establishes the Euclidean-Sphere correspondence of optimal partitions via stereographic projection.

Abstract

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

Fractional $Q$-curvature on the sphere and optimal partitions

TL;DR

This work studies a fractional Yamabe-type problem on the sphere, where the cost is governed by the fractional Q-curvature via the conformal fractional Laplacian . The authors exploit -symmetry to recover compactness in a critical nonlocal setting and prove the existence of a symmetric minimal partition of the sphere, with the partition profiles arising as phase-separated limits of a weakly coupled fractional system. A key methodological innovation is a self-contained, symmetry-driven reduction to a one-dimensional setting using Jacobi polynomials, which yields a new Hölder regularity result for symmetric -functions on the sphere (for ). The work further demonstrates the existence of infinitely many symmetric, fully nontrivial solutions to nonlocal competitive systems on the sphere and in Euclidean space, together with a precise description of the asymptotic phase-separation behavior as coupling tends to minus infinity, and establishes the Euclidean-Sphere correspondence of optimal partitions via stereographic projection.

Abstract

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional -curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

Paper Structure

This paper contains 26 sections, 35 theorems, 175 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The fractional $Q$-curvature problem.
  3. The conformal fractional Laplacian on the sphere.
  4. The optimal partition problem.
  5. A proof via phase separation.
  6. Seizing the one-dimensional structure
  7. Fractional Laplacians
  8. The spectrum of the conformal fractional Laplacian on the sphere
  9. Spaces with symmetries
  10. Jacobi Polynomials
  11. One-dimensional weighted spaces
  12. Regularity properties of fractional Sobolev spaces with symmetries
  13. Further properties of Jacobi polynomials
  14. Jacobi polynomials and regularity
  15. Variational problems with symmetries
  16. ...and 11 more sections

Key Result

Lemma 1.1

For $u\in C^\infty(\mathbb S^N)$, $v := u\circ \sigma^{-1}$, $z\in \mathbb S^N$, and $x=\sigma(z) \in \mathbb R^N$, it holds that

Figures (1)

  • Figure 1: Illustration of the optimal partition $\{\Omega_1,\Omega_2,\Omega_3,\Omega_4\}$ of $\mathbb{R}^3$ given by Theorem \ref{['thm:main_RNPart1']} with $\ell=4$ and $m=n=2$. Here $\Omega_1$ is the interior of the smallest torus, $\Omega_4$ is the exterior of the largest torus, and $\Omega_2$ and $\Omega_3$ are the spaces inbetween separated by the middle torus.

Theorems & Definitions (69)

  • Lemma 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 59 more